A Recentered-Domain Yau–Yau Filter with Reduced-Order FKE Propagation for Nonlinear State Estimation

Nonlinear filtering can be formulated as the propagation and update of conditional probability densities, but direct numerical propagation of the associated Forward Kolmogorov equation (FKE) over a large fixed domain is computationally expensive. This paper proposes a Recentered-Domain Yau–Yau Filter (RD-YYF) with reduced-order FKE propagation for nonlinear state estimation. The method solves the FKE on a fixed-size local computational window centered at the latest state estimate, thereby concentrating numerical resolution near the dominant posterior density. In the offline stage, physics-informed neural networks (PINNs) generate FKE solution snapshots, principal component analysis constructs a low-dimensional representation of density evolution, and a lightweight residual surrogate maps initial-condition coefficients and the domain center to terminal-solution coefficients. In the online stage, the pretrained surrogate performs per-timestep density prediction within the recentered window, followed by observation update and state estimation. Numerical experiments on two geometrically constrained target-tracking models show that RD-YYF achieves lower tracking errors than the extended Kalman filter and particle filter under matched online evaluation conditions. A fixed-domain ablation further shows that recentering improves density approximation in high-probability regions and reduces offline PINN training epochs. These results indicate that recentered-domain reduced-order FKE propagation is a practical computational strategy for nonlinear density-based filtering.